All Questions
Tagged with colimits or limits-and-colimits
347 questions
6
votes
0
answers
152
views
Covering categories with posets
Let $C$ be a small (1-)category.
There is always a poset $D$ and a functor $p : D \to C$ such that:
$p$ is surjective on objects, i.e. for every $c$ in $C$ there is a $d$ in $D$ such that $p (d) = c$,...
- 15.9k
5
votes
1
answer
198
views
Density formula in small presheaves
I've been trying to write down a proof of Di Liberti's Kan lemma fortissimo on the existence of left Kan extensions, given as Lemma 3.3 in the nLab entry on Kan extensions. Let $F : \mathcal{A} \to \...
- 51
2
votes
1
answer
74
views
Conditions for partially applied induced product functor to preserve colimits
Let $\mathcal{C}$ have products, $A, B \in \mathcal{C}_0$, ${\times}\colon \mathcal{C}^2\to \mathcal{C}$ be the functor that sends objects to their product.
Then the induced product ${\boxtimes}\colon ...
- 25
1
vote
2
answers
216
views
Presentable categories as colimits of finitely presentable categories
I am trying to understand the relationship betweeen compactly generated presentable categories, also called finitely presentable categories, and general presentable categories (which I have less ...
- 521
5
votes
1
answer
156
views
The inverse limit of a sequence of ring surjections commutes with taking difference subsets of the respective units & gluing in some primes?
Define $R_n := \Bbb{Z}/p_n\#$ the ring of integers modulo primorial $p_n\# = p_n p_{n-1} \cdots p_1$. Let $U_n$ denote the group of units modulo $p_n\#$ in these rings.
Then if $f_{n,n+1}: \Bbb{Z}/p_{...
- 260
5
votes
0
answers
361
views
On a simple alternative correction to Roos' theorem on $\varprojlim^1$
Here is a discussion about an incorrect theorem of Roos, later corrected, some counterexamples and so on. Reading over this, I was a bit shocked because it contradicted something from Weibel's ...
- 1,021
11
votes
1
answer
451
views
Is every petite category essentially small?
A locally small category $\mathscr C$ is called petite if, for every functor $F : \mathscr C \to \mathscr D$ with locally small codomain, and for every object $D \in \mathscr D$, the presheaf $\...
- 10.6k
1
vote
1
answer
234
views
Notion of $\kappa$-sifted categories?
Let $\kappa$ be a regular cardinal. It seems reasonable to introduce the following definition:
Definition. A simplicial set $K$ is $\kappa$-sifted if, for every set $E$ with $\lvert E\rvert<\kappa$...
- 2,806
8
votes
1
answer
390
views
Pushouts vs contractible colimits
Suppose that $C$ has all weakly contractible colimits, i.e. colimits of functors $F: I \rightarrow C$ where the geometric realization $|I|$ is weakly contractible. Then $C$ has pushouts and filtered ...
- 521
3
votes
0
answers
55
views
Universal property of 2-presheaves and pseudo/lax/colax natural transformations
For each small 2-category $\mathscr K$, the 2-category $[\mathscr K^\circ, \mathrm{Cat}]$ of 2-functors and 2-natural transformations has a universal property: it is the free cocompletion of $\mathscr ...
- 10.6k
5
votes
0
answers
89
views
Free cocompletion of a 2-category under pseudo colimits, lax colimits, and colax colimits
Let $\mathscr K$ be a small 2-category. It follows from $\mathrm{Cat}$-enriched category theory that the free cocompletion of $\mathscr K$ under strict 2-colimits of 2-functors is given by the 2-...
- 10.6k
6
votes
1
answer
281
views
Are adjoints closed under pushouts?
The category $PrL$ of locally presentable categories has all colimits. In particular, if
$A_1 \leftarrow A_0 \rightarrow A_2$ is a diagram of presentable categories, with left adjoint functors between ...
- 521
1
vote
0
answers
54
views
contravariant finiteness and limit closure: is there dual to a result of Crawley-Boevey?
Let $\mathcal A$ be a locally finitely presented category. Theorem 4.2 of https://doi.org/10.1080/00927879408824927 says that given a full additive subcategory $\mathcal D$ of finitely presented ...
- 480
4
votes
1
answer
140
views
Cocompletion without cocontinuous functors
The forgetful functor from the 2-category $\mathsf{Cats}^{\mathrm{loc.small}}_{\mathrm{cocomp}}$ of locally small cocomplete categories and cocontinuous functors to the 2-category $\mathsf{Cats}^{\...
- 11.8k
5
votes
0
answers
155
views
Compact objects in categories of categories
I am interested in the compact objects of various categories of categories.
For example, $Cat^{small}$ is presentable and has compact objects that are retracts of finite colimits of $\Delta^n$, the $n$...
- 521
1
vote
0
answers
127
views
Extremally disconnected sets as building blocks for compact Hausdorff spaces
Is every compact Hausdorff space the filtered colimit of compact extremally disconnected spaces?
- 1,638
4
votes
2
answers
314
views
Directed colimit of fully faithful functors
Suppose that for every $n\in\mathbb{N}$ we have a category $\mathcal{C}_n$ and a fully faithful functor $F_n:\mathcal{C}_n\hookrightarrow \mathcal{C}_{n+1}$. My question is whether fully faithful ...
- 119
5
votes
1
answer
436
views
Does coproduct preserve cohomology in differential graded algebra category
Consider two cochain DGA (differential graded algebras) named $A$ and $B$. By "coproduct" of two DGA I mean the category theory coproduct, not the coalgebra coproduct. It is defined in "...
- 159
-1
votes
1
answer
178
views
Categories that admit all finite products but not all finite coproducts
What are examples for categories that admit all finite products but not all finite coproducts?
(See also this question: Categories that admit all products but not all coproducts .)
5
votes
1
answer
158
views
Co/limits and 2-co/limits of categories in the $\infty$- and $(\infty,2)$-category of $\infty$-categories
Recently, in a conversation with Gabriel, the following question came up:
Question. Do co/limits of categories taken in the $\infty$-category of $\infty$-categories agree with the usual co/limits ...
- 11.8k
2
votes
0
answers
100
views
Weighted limits and co-Yoneda
Is there a good reference that discusses weighted limits through the lens of the co-Yoneda embedding?
Recall that the limit of a functor $F:\mathcal{C}\to{\bf Set}$ is canonically given by the set $${...
- 10.1k
13
votes
1
answer
224
views
Large V-categories admitting the construction of V-presheaves
By a result of Foltz, and Freyd and Street, a category $C$ is essentially small (i.e. equivalent to a small category) if and only if both $C$ and $[C^{\text{op}}, \mathrm{Set}]$ are locally small. I ...
- 10.6k
1
vote
0
answers
88
views
Colimits from van Kampen cocones
Let $\mathcal{C}$ be a category with pullbacks, $\mathcal{J}$ a small category, $F : \mathcal{J} \to \mathcal{C}$ a diagram and $\kappa : F \Rightarrow \Delta X$ a cocone in $\mathcal{C}$. Let $\...
- 453
3
votes
1
answer
223
views
Kernels and cokernels in a quotient of an abelian category
I am trying to understand the construction of the quotient of an abelian category called the Serre quotient or Gabriel quotient. From the description here: https://en.wikipedia.org/wiki/...
- 323
8
votes
1
answer
253
views
Compact objects in slice categories of finitely presentable categories
Given a locally finitely presentable category $\mathscr C$ and an object $X \in \mathscr C$, it is not so hard to show that a morphism $(X \to Y)$ is compact in $\mathscr C_{X/}$ if it can be obtained ...
- 28.7k
5
votes
1
answer
411
views
Day convolution and sheafification
$\DeclareMathOperator\Psh{Psh}\DeclareMathOperator\Sh{Sh}\newcommand\copower{\mathrm{copower}}$I was looking through Bodil Biering's thesis On the Logic of Bunched Implications - and its relation to ...
6
votes
1
answer
155
views
Does the 2-category of double categories and vertical transformations have flexible limits?
Consider the 2-category of pseudo-double categories (with the weak composition in the horizontal direction and the strict composition in the vertical direction), strong double functors, and vertical ...
7
votes
1
answer
253
views
Can every weighted colimit in a $\mathbf{Pos}$-enriched category be rephrased as a conical colimit?
For ordinary category theory, we have the following fact.
A weighted colimit of a functor can always be equivalently expressed as a colimit of a different functor.
Specifically, the weighted colimit ...
- 173
6
votes
1
answer
233
views
Comparing stabilization of stable category modulo injectives and a Verdier localization
Let $\mathcal A$ be an abelian category with enough injectives. Let $\mathcal I$ be the collection of injective objects. Let $\mathcal A/\mathcal I$ be the quotient category whose objects are same as ...
- 413
3
votes
1
answer
123
views
Vanishing of self-hom in Spanier–Whitehead stabilization category
$\DeclareMathOperator\Hom{Hom}\DeclareMathOperator\SW{SW}$Let $R$ be a commutative Noetherian ring. For $R$-modules $M,N$, let $\mathcal I_R(M,N)$ be the collection of all $f\in \text{Hom}_R(M,N)$ ...
- 413
8
votes
1
answer
258
views
Sequential colimit of iterated quotients of Cauchy sequences
We work in constructive mathematics.
The sets and functions in the foundations form a Grothendieck topos, which means that all colimits exist, and in particular, that all sequential colimits exist. ...
- 2,624
5
votes
0
answers
245
views
Duality and compactness for pro vector spaces
I have a somewhat basic question which I haven't been able to piece together from the literature.
Background. We work over a field $\bf{k}$. Consider the category, $\bf{Pro}_{k}$, of pro- vector ...
- 51
9
votes
2
answers
421
views
How do these definitions of factorization algebra compare?
Question
Several sources define (homotopy) factorization algebras in a seemingly
different manner (I am looking at [CG], [Gi], and
[CFM].) I wish to know how they compare with each other.
I apologize ...
- 2,292
4
votes
0
answers
131
views
Weakening of terminal object in a category
I’ve come across a category $\mathcal{C}$ recently with an object $T$ such that any other object $X$ has a map $f:X\rightarrow T$, and for any two maps $f,g:X\rightarrow T$, there exists a (not ...
- 1,949
9
votes
0
answers
103
views
Adjoining a morphism to a finitely complete category
Let $\mathscr C$ be a finitely complete category. Let $x, y$ be objects of $\mathscr C$. We can describe the universal property of freely adjoining a morphism $x \to y$ to $\mathscr C$: it comprises a ...
- 10.6k
6
votes
1
answer
315
views
Commuting homotopy colimits and arbitrary products in spaces
Let $X : D \rightarrow Spc$ be a diagram with values in the $\infty$-category of spaces and $I$ some (discrete) set, not necessarily finite. ($D$ can be a 1-category if that makes statements easier, ...
- 2,303
2
votes
1
answer
168
views
Inclusion of $1$-presheaves into $\infty$-presheaves preserves pushouts?
Let $\mathcal{R}$ be a $1$-category. Assume that one has a pushout of representable $1$-presheaves $\mathrm{y} A \cup_{\mathrm{y} B} \mathrm{y} C$ in $\mathsf{PSh}(\mathcal{R})$. Under which ...
- 355
6
votes
1
answer
480
views
Why isn't pullback-stability defined for individual colimits but for colimits with the same shape?
Circumstances: I'm studying Grothendieck's Galois Theory and recently encountered a proposition that discussed the stability of coproducts under pullback. And I found the page pullback-stable colimit ...
- 165
6
votes
2
answers
425
views
Has the "Lambek embedding" into the category of (co)product-preserving presheaves been studied very much?
The Lambek embedding is a particular embedding which is similar to the Yoneda embedding.
Suppose we have any category $C$. Recall that a presheaf on $C$ is defined as a contravariant functor from $C$ ...
- 1,173
10
votes
1
answer
498
views
What does it mean for a category to be generated under (some) colimits?
This is going to be a long post, so let me state my question first and then explain what I am interested in by way of examples.
Question.
Is there any literature studying notions of generation under ...
- 15.9k
1
vote
1
answer
232
views
Pointwise Kan extensions VS weighted limits
$\newcommand{\Dist}{\operatorname{Dist}} \newcommand{\Ran}{\operatorname{Ran}} \newcommand{\Lim}{\operatorname{Lim}}$
TLDR
Given a pointwise kan extension, how can we go from
$$ \Dist(B, C)(\phi_c \...
- 231
3
votes
1
answer
147
views
Does the forgetful functor from a pointed $\left(\infty, 1\right)$-category only create weakly contractible colimits?
Consider an $\left(\infty, 1\right)$-category $\mathcal{C}$ with a terminal object $1$. (I'm particularly interested in the case where $\mathcal{C}$ is a topos.) It is known that the forgetful functor ...
- 81
2
votes
1
answer
95
views
Weighted limits and Kan extension in Dist
(noting $\otimes$ for composition in distributors, $\phi_f : A \nrightarrow B = B(-,f=)$ and $\phi^f : B \nrightarrow A = B(f-,=)$ the embeddings of a functor $f:A\to B$ in $Dist$, and $Dist(A,B) = [B^...
- 231
6
votes
1
answer
261
views
When does base-change in topological spaces preserve quotient maps?
The question when $(-) \times X$ preserves colimits in topological spaces is well-studied. Since it always preserves arbitrary coproducts (disjoint unions), one only has to show when it preserves ...
- 12.1k
5
votes
1
answer
465
views
Homotopy groups of categories of elements as higher colimits
Given a diagram of sets $D\colon\mathcal{C}\to\mathsf{Set}$, we have a bijection (Proof)
$$\operatorname{colim}(D) \cong \pi_0 (\textstyle\int_\mathcal{C}D).$$
Is there any known application or ...
- 11.8k
9
votes
0
answers
129
views
Is totality a (large) cocompleteness condition?
A locally small category $A$ is called total if its Yoneda embedding $A \to [A^\circ, \mathbf{Set}]$ has a left adjoint. Such categories are necessarily small-cocomplete (since the presheaf category ...
- 10.6k
2
votes
0
answers
163
views
Initial cones, terminal cocones
We're all familiar with terminal cones/initial cocones in the form of limits/colimits.
What about initial cones and terminal cocones?
While writing an answer to a related question the concept ...
- 10.1k
2
votes
1
answer
180
views
Non-cofiltered derived limits
As far as I know, the inverse limit and its derived functors can be defined even in case we are dealing with a functor $F: I \to A$ from a category $I$ that is not cofiltered. I would content myself ...
- 505
6
votes
0
answers
142
views
Which Ends preserve filtered colimits?
Can we characterize entirely for which categories $C$ the end on $C$ preserves filtered colimits, in a sense that the natural map
$$ \operatorname*{colim}_{i \in I} \int_{x \in C} A_i (x,x) \to \int_{...
- 42.4k
3
votes
1
answer
619
views
Homotopy colimit commutes with homotopy groups
I'm interested in something built upon the construction laid out in nlab article on Snaith's theorem
Let $(E, \mu, \iota)$ be a ring spectrum.
For $\beta \in \pi_n(E)$ an element of the $n$th stable ...
- 301